SOLIDS ON SURFACES. 349 



ble; the formulas I have given will however be very useful 

 in investigating the position of the body. If there be abso- 

 lutely no friction, in that case the above equations of condition 



are the only ones which exist along with the general dynami- 

 cal equation, lint if there be proposed any hypothesis of frie- 

 tion or analogous restraint, the following considerations will 

 assist us in determining the relations between the momentary 

 changes in translation and rotation. 



Let us. for the sake of greater generality, suppose that the 

 two bodies J/" and M . which are in contact with each other, 

 are both of them in motion. There will be now at least six 

 different velocities at the point of contact, liable without atten- 

 tion to be confounded with each other: — I. The absolute ve- 

 locity in space of the physical point of contact ]> of the body 

 M. II. The absolute velocity in space of the physical point 

 of contact jo belonging to the body M . III. The absolute 

 velocity of the geometrical point of contact P. IV. The 

 velocity with which the point P changes its place on the sur- 

 face of M. V. The velocity with which the same point /' 

 changes its place on the surface of the body M . VI. The 

 velocity of rasure. — The same distinctions are to be observed 

 with respect to the directions which belong to these velocities. 

 The effects of friction at the point of contact will depend en- 

 tirely upon the velocity and direction of rasure, which are the 

 same with which the physical points p and j> recede from 

 each other in the instant after they meet at the geometrical 

 point /'. If one of the bodies as M he li\ed. then this velo- 

 city and direction will be the same with the absolute velocity 



and direction of the physical point p. and the velocities of ra- 

 siire in the direction of the bodj coordinates will therefore 

 be denoted generally by 



,/c -ydJR-{-Z,dQ, 



,l r _ r ,//>_ H .,■,//,'. 



or. when necessary, by the values (g }). which we have shewn 

 to be equivalent to the above expressions. We maj suppus, 



